Question

solve this 50 points

if a cone of height 8m jas a curved surface area of 188.4m then find the radius of its base and find its volume take (π=3.14)​

Answer 1

† Question †

if a cone of height 8m jas a curved surface area of 188.4m then find the radius of its base and find its volume take (π=3.14)

* Given

→ h=8m

→ c.s.a=188.4m

→ r = ?

→ v = ?

$\rule{300}2$

【 solution 】

$l^2=r^2+h^2 \\ l^2=r^2+8^2\\l^2=r^2+64\\l=\sqrt{r^2+64}$

c.s.a=πrl

$=188.4=3.14×r×\sqrt{r^2+64} \\=r×\sqrt{r^2+64}=60$

squaring both side

$=r^2×(r^2+64)=3600\\=r^4+65r^2+3600=0$

$let = r^2=y \\y^2+64y-3600=0\\y^2+(100-36)y-3600=0\\y^2+100h-36y-3600=0\\y(y+100)-36(y+100)=0\\(y+100)(y-36)=0\\y+100=0,y=-100 \\y-36=0\\y=36 \\r^2=36\\r=6m\\v=\frac{1}{3}πr^2h\\\frac{1}{3}×3.14×6×6×h\\3.14×12h\\h=3.14×12\\h=37.68cm^3$

*awesome question*

Answer 2

Answer:

${ \huge{ \bf{ \underline{ \red{Given}}}}}$

Height of cone = 8m

C.S.A of cone = 188.4m

Find:-

Volume of cone

Solution:-

C.S.A of cone = πrl

So, now we have to find value of L :-

${ \to{ \sf{ {</u></strong><strong><u>L</u></strong><strong><u>}^{2} = {r}^{2} + {h}^{2} }}}$

${ \to{ \sf{ {</u></strong><strong><u>L</u></strong><strong><u>}^{2} = {r}^{2} + {8}^{2} }}}$

${ \to{ \sf{ {</u></strong><strong><u>L</u></strong><strong><u>}</u></strong><strong><u>^{2} = {r}^{2} + 64}}}$

${ \to{ \sf{</u></strong><strong><u>L</u></strong><strong><u> = \sqrt{ {r}^{2} + 64} }}}$

C.S.A of cone = πrl

${ \sf{188.4 = 3.14 \times r \times \sqrt{ {r}^{2} + 64} }}$

${ \sf{ \frac{188.4}{3.14} = r \times \sqrt{ {r}^{2} + 64} }}$

${ \sf{60 = r \times \sqrt{ {r}^{2} + 64} }}$

DO S.O.B.S:-

${ \sf{ {60}^{2} = {r}^{2} + { \sqrt{ {r}^{2} + 64 } }^{2} }}$

${ \sf{3600 = {r}^{2} × {r}^{2} + 64 }}$

${ \sf{ {r}^{4} + 64 {r}^{2} - 3600 = 0 }}$

${ \sf{ {r}^{2} + 64 {r}^{2} - 3600 = 0 }}$

${ \sf{ {r}^{2} + 100x - 36x - 3600 = 0}}$

${ \to{ \sf{x(x + 100) - 36(x + 100) }}}$

(x+100) (x-36) =0

x+100=0 = >x = -100

x-36 = 0

x= 36

${ \sf{ {r}^{2} = 36 }}$

${ \sf{r = \sqrt{36} }}$

${ \sf{r = 6}}$

Volume of cone = 1/3πr²h

${ \to{ \sf{ \frac{1}{3} \times 3.14 \times {6}^{2} \times 8}}}$

${ \to{ \sf{ \frac{1}{3 } \times 3.14 \times 36 \times 8 }}}$

${ \to{ \sf{ \frac{1}{3} \times 3.14 \times 288}}}$

${ \to{ \sf{ \frac{1}{3} \times 904.32}}}$

${ \to{ \sf{ \frac{904.32}{3} }}}$

${ \to{ \sf{301.44}}}$

${ \therefore { \sf{volume \: of \: cone \: = 301.44}}}$